$3.$ Find the close form complexity function of T$($n$)=3$T$($n$-1)+1,$

while T$(1)=1(10$ points$)$

$4.$ Suppose you have two different algorithms with the following running times to solve

the same problem. $(10$ points$)$

For algorithm$1$: T$($n$)=(104)$n $+100$

For algorithm$2$: T$($n$)=2($n$2)+20$n $+100$

When will algorithm$1$ outperform algorithm$2$ and vice versa? Explain your answer.

$5.$ For the following pseudo$-$code, what is the time complexity function T$($n$)$

and the order $($O$)?$ You can ignore the overhead operations and just count the

basic operations. $(30$ points$)$

for $($i$=1$; i$<=$n; i$++)$

for $($j$=1$; j$<=3*$i; j$++)$

for $($k$=1$; k$<=3*$n; k$++)$

cout $<<"*"$ ;